Math 641-600 — Fall 2019

Assignment 12 - Due Monday, December 2, 2019.

• Read sections 4.1, 4.2, 4.3.1, 4.3.2, 4.5.1 and my notes on and my notes on example problems for distributions.
• Do the following problems.
  1. Section 3.5: 2(b)

  2. Section 4.1: 4, 7

  3. Section 4.2: 1, 3, 4

  4. Section 4.3: 3

  5. Let $F:C[0,1]\to C[0,1]$ be defined by $F[u](t) := \int_0^1(2+st+u(s)^2)^{-1}ds$, $0\le t\le 1$. Let $\| \cdot \|:=\|\cdot \|_{C[0,1]}$. Let $B_r:=\{u\in C[0,1]\,|\, \|u\|\le r\}$.
     1. Show that $F: B_1\to B_{1/2}\subset B_1$.
     2. Show that $F$ is Lipschitz continuous on $B_1$, with Lipschitz constant $0<\alpha \le 1/2$.
     3. Show that $F$ has a fixed point in $B_1$.

  6. Let $Ku(x)=\int_0^1 k(x,y)u(y)dy$, where $k(x,y)$ is defined by $ k(x,y) = \left\{ \begin{array}{cl} y, & 0 \le y \le x\le 1, \\ x, & x \le y \le 1. \end{array} \right.$
     1. Show that $0$ is not an eigenvalue of $K$.
     2. Show that $Ku(0)=0$ and $(Ku)'(1)=0$.
     3. Find the eigenvalues and eigenvectors of $K$. Explain why the (normalized) eigenvectors of $K$ are a complete orthonormal basis for $L^2[0,1]$.

  7. Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
     1. Show that the Green's function for this problem is \[ G(x,y)=\left\{ \begin{array}{rl} -(2y-1)x, & 0 \le x < y \le 1\\ -(2x-1)y, & 0 \le y< x \le 1. \end{array} \right. \]
     2. Let $Kf(x) := \int_0^1G(x,y)f(y)dy$. Show that $K$ is a self-adjoint Hilbert-Schmidt operator, and that $0$ is not an eigenvalue of $K$.
     3. Use (b) and the spectral theory of compact operators to show the orthonormal set of eigenfunctions for $L$ form a complete set in $L^2[0,1]$.

Updated 11/22/2019.